Statistical law observed in inactive rate of firms

In this study, we observe the dependence of the inactive rate of firms on firm-size variables: total revenue, net income, total assets, and net assets. This is achieved using information on nearly all firms in Japan, Germany, Spain, France, the United Kingdom, Italy, Korea, and the Netherlands recorded in the Orbis database in 2015 and 2016. Here, the inactive rate means the transition rate of firms from an active state to an inactive state. First, we confirm that the financial variables in this database follow power laws in the range of large-scale data values but follow log-normal distributions in the range of mid-scale data values. Furthermore, we observe that the inactive rate of firms is constant regardless of the firm-size variables in the large-scale data range. Meanwhile, the inactive rate increases under a power-law function as financial variables decrease in the mid-scale data range. We also find that the boundary between the large- and mid-scale data ranges of the inactive rate of firms corresponds to the boundary between the power-law and log-normal distributions of the financial variables.